Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#32549 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C4×Q8×C13 ◁ G | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 3328 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,18)(16,20,21,19)(22,23)(24,28)(25,27)(26,29);(15,18)(16,21)(22,24)(23,26)(25,29)(27,28)(30,31,35,38)(32,39,41,44)(33,37,34,42)(36,43,45,40);(23,27)(26,28)(30,32,35,41)(31,36,38,45)(33,40,34,43)(37,44,42,39);(14,16,17,21)(15,19,18,20)(24,29)(26,28)(30,33,35,34)(31,37,38,42)(32,40,41,43)(36,44,45,39)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 25537);